In this paper, we characterize all N-dimensional hypercomplex numbers having unital Archimedean f-algebra structure. We use matrix representation of hypercomplex numbers to define an order structure on the matrix spectra. We prove that the unique (up to isomorphism) unital Archimedean f-algebra of hypercomplex numbers of dimension \(N \ge 1\) is that with real and simple spectrum. We also show that these number systems can be made into unital Banach lattice algebras and we establish some of their properties. Furthermore, we prove that every 2N-dimensional unital Archimedean f-algebra is the hyperbolization of that of dimension N. Finally, we consider hypercomplex number systems of dimension \(N=2,3,4,6\) and give their explicit matrix representation and eigenvalue operators. This work is a multidimensional generalization of the results obtained in Gargoubi and Kossentini (Adv Appl Clifford Algebras 26(4):1211–1233, 2016) and Bilgin and Ersoy S (Adv Appl Clifford Algebras 30:13, 2020) for, respectively, the two and four-dimensional systems.

在本文中，我们描述了所有具有单位阿基米德f代数结构的N维超复数。我们使用超复数的矩阵表示来定义矩阵谱上的有序结构。我们证明了\(N \ge 1\)维超复数的唯一（直到同构）单位阿基米德f代数是具有实数和简单谱的。我们还表明，这些数系可以制成单位巴纳赫格代数，并建立了它们的一些性质。进一步证明，每一个2 N维单位阿基米德f代数都是N维单位阿基米德f 代数的双曲线。最后，我们考虑维数\(N=2,3,4,6\)的超复数系统，并给出它们的显式矩阵表示和特征值算子。这项工作是 Gargoubi 和 Kossentini (Adv Appl Clifford Algebras 26(4):1211–1233, 2016) 以及 Bilgin 和 Ersoy S (Adv Appl Clifford Algebras 30:13, 2020) 分别获得的结果的多维推广，分别为：二维和四维系统。